You are using the "Basic Authflow" from cognito identity, which means you will need to get credentials for your users by calling STS's "AssumeRoleWithWebIdentity". Here is some documentation to help: http://docs.aws.amazon.com/cognito/devguide/identity/concepts/authentication-flow/
Once you have credentials, you can instantiate the API Gateway Client:
var client = apigClientFactory.newClient({
accessKey: ACCESS_KEY,
secretKey: SECRET_KEY,
sessionToken: SESSION_TOKEN });
The keys and tokens come from the result of the "AssumeRoleWithWebIdentity" call.
If you have configured your IAM roles, and Authorizations correctly you should be able to access your API.
Here is the documentation describing how to configure the roles & authorization: http://docs.aws.amazon.com/apigateway/latest/developerguide/how-to-method-settings.html#how-to-method-settings-callers-console
Also, here is how to enable CORS - http://docs.aws.amazon.com/apigateway/latest/developerguide/how-to-cors.html